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Some monotonicity properties and inequalities for the generalized digamma and polygamma functions

Abstract

Several monotonicity and concavity results related to the generalized digamma and polygamma functions are presented. This extends and generalizes the main results of Qi and Guo and others.

Introduction

The Euler gamma function is defined for all positive real numbers x by

$$ \Gamma(x)= \int_{0}^{\infty}t^{x-1}e^{-t}\,dt. $$

The logarithmic derivative of \(\Gamma(x)\) is called the psi or digamma function. That is,

$$ \psi(x)=\frac{d}{dx}\ln\Gamma(x)=\frac{\Gamma'(x)}{\Gamma (x)}=-\gamma - \frac{1}{x}+\sum_{n=1}^{\infty} \frac{x}{n(n+x)}, $$

where \(\gamma=0.5772\ldots\) is the Euler–Mascheroni constant, and \(\psi^{(m)}(x)\) for \(m\in\mathbb{N}\) are known as the polygamma functions. The gamma, digamma and polygamma functions play an important role in the theory of special functions, and have many applications in other many branches, such as statistics, fractional differential equations, mathematical physics and theory of infinite series. The reader may see the references [913, 1820, 24, 4547, 49]. Some of the work on the complete monotonicity, convexity and concavity, and inequalities of these special functions can be found in [16, 8, 1417, 21, 22, 2730, 3742] and the references therein.

In 2007, Diaz and Pariguan [11] defined the k-analogue of the gamma function for \(k>0\) and \(x>0\) as

$$ \Gamma_{k}(x)= \int_{0}^{\infty}t^{x-1}e^{-\frac{t^{k}}{k}}\,dt= \lim_{n\rightarrow\infty}\frac{n!k^{n}(nk)^{\frac{x}{k}-1}}{x(x+k)\cdots (x+(n-1)k)}, $$

where \(\lim_{k\rightarrow1}\Gamma_{k}(x)=\Gamma(x)\). Similarly, we may define the k-analogue of the digamma and polygamma functions as

$$\psi_{k}(x)=\frac{d}{dx}\ln\Gamma_{k}(x) \quad \mbox{and} \quad \psi _{k}^{(m)}(x)=\frac {d^{m}}{dx^{m}} \psi_{k}(x). $$

It is well known that the k-analogues of the digamma and polygamma functions satisfy the following recursive formula and series identities (see [11]):

$$\begin{aligned}& \Gamma_{k}(x+k)=x\Gamma_{k}(x),\quad x>0, \end{aligned}$$
(1.1)
$$\begin{aligned}& \psi_{k}(x)=\frac{\ln k-\gamma}{k}-\frac{1}{x}+\sum _{n=1}^{\infty }\frac{x}{nk(nk+x)}, \end{aligned}$$
(1.2)

and

$$ \psi_{k}^{(m)}(x)=(-1)^{m+1}m!\sum _{n=0}^{\infty}\frac{1}{(nk+x)^{m+1}}. $$
(1.3)

Very recently, Nantomah, Prempeh and Twum [35] introduced a \((p,k)\)-analogue of the gamma and digamma functions defined for \(p\in\mathbb{N}\), \(k>0\) and \(x>0\) as

$$\begin{aligned}& \Gamma_{p,k}(x)= \int_{0}^{p} t^{x-1} \biggl(1- \frac{t^{k}}{pk} \biggr)^{p}\,dt=\frac {(p+1)!k^{p+1}(pk)^{\frac{x}{k}-1}}{x(x+k)\cdots(x+pk)}, \end{aligned}$$
(1.4)
$$\begin{aligned}& \psi_{p,k}(x)=\frac{d}{dx}\ln\Gamma_{p,k}(x)= \frac{1}{k}\ln (pk)-\sum_{n=0}^{p} \frac{1}{nk+x}, \end{aligned}$$
(1.5)

and

$$\begin{aligned} \psi_{p,k}^{(m)} (x) =& ( - 1){}^{m}m!\sum _{n = 0}^{p} {\frac{1}{{(nk + x)^{m + 1} }}} \\ =& ( - 1)^{m + 1} \int_{0}^{\infty}{\frac{{1 - e^{ - k(p + 1)t} }}{{1 - e^{ - kt} }}} t^{m} e^{ - xt} \,dt. \end{aligned}$$
(1.6)

It is obvious that \(\lim_{p\rightarrow+\infty}\psi_{p,k}(x)=\psi_{k}(x)\). Some important identities and inequalities involving these functions may be found in [30, 34, 35].

In [4], the function \(\phi(x)=\psi(x)+\ln(e^{\frac {1}{x}}-1)\) was proved to be strictly increasing on \((0,\infty)\). In [6], it is demonstrated that if \(a\leq-\gamma\) and \(b\geq0\), then

$$ a-\ln\bigl(e^{\frac{1}{x}}-1\bigr)< \psi(x)< b-\ln \bigl(e^{\frac{1}{x}}-1\bigr). $$
(1.7)

Furthermore, Guo and Qi [14] showed that the function \(\phi(x)\) is strictly increasing and concave on \((0,\infty)\). Attracted by this work, it is natural to look for an extension of (1.7) involving \(\psi_{k}(x)\) and \(\psi_{p,k}(x)\). On the other hand, Nielsen’s β-function has been deeply researched in the last years. In particular, K. Nantomah gave some results on convexity and monotonicity of the function in [31], and obtained some convexity and monotonicity results as well as inequalities involving a generalized form of the Wallis’s cosine formula in [32]. The function can be used to calculate some integrals (see [7, 36]). Recently, K. Nantomah studied the properties and inequalities of a p-generalization of the Nielsen’s function in [33]. In this paper, we shall give double inequalities for the k-generalization of the Nielsen β-function. In addition, it is worth noting that Krasniqi, Mansour, and Shabani presented some inequalities for q-polygamma functions and q-Riemann Zeta functions by using a q-analogue of Hölder type inequality in [23].

The first aim of this paper is to present a new monotonicity theorem for \(\psi_{k}(x)\), and give three different proofs. The second aim is to show an inequality for the ratio of the generalized polygamma functions by generalizing a method of Mehrez and Sitnik. The classical Mehrez and Sitnik’s method may be found in [25, 26, 43]. Finally, we also give a new inequality for the inverse of the generalized digamma function.

Our main results read as follows.

Theorem 1.1

For \(0< k\leq1\), the function \(\phi_{k}(x)=\psi_{k}(x)+\ln(e^{\frac {1}{x}}-1)\) is strictly increasing on \((0,\infty)\). In particular, the inequalities

$$ \frac{\ln k-\gamma}{k}< \psi_{k}(x)+\ln\bigl(e^{\frac{1}{x}}-1 \bigr)< 0 $$
(1.8)

hold true for \(0< k\leq1\) and \(x\in(0,\infty)\) where the constants \(\frac{\ln k-\gamma}{k}\) and 0 in (1.8) are the best possible.

Remark 1.1

Here, we give an application of Theorem 1.1. Define the k-generalization of the Nielsen’s β-function as

$$\begin{aligned} \beta_{k}(x)&= \int_{0}^{1} \frac{t^{x-1}}{1+t^{k}}\,dt \\ &= \int_{0}^{\infty} \frac{e^{-xt}}{1+e^{-kt}}\,dt \\ &=\sum^{\infty}_{n=0} \biggl( \frac{1}{2nk+x}-\frac{1}{2nk+k+x} \biggr) \\ &=\frac{1}{2} \biggl\{ \psi_{k} \biggl(\frac{x+k}{2} \biggr)-\psi_{k} \biggl(\frac{x}{2} \biggr) \biggr\} . \end{aligned}$$

By using (1.8), we easily obtain double inequalities of the generalized Nielsen’s β-function for \(0< k\leq1\) and \(x\in (0,\infty)\):

$$\frac{1}{2}\ln \biggl( {\frac{{e^{2/x} - 1}}{{e^{2/(x + k)} - 1}}} \biggr) + \frac{{\ln k - \gamma}}{2k} < \beta_{k} (x) < \frac{1}{2}\ln \biggl( {\frac{{e^{2/x} - 1}}{{e^{2/(x + k)} - 1}}} \biggr) - \frac{{\ln k - \gamma}}{2k}. $$

Theorem 1.2

For \(0< k\leq1\), the function \(\phi_{k}(x)\) is strictly concave on \((0,\infty)\). As a result, for \(0< k\leq1\) and \(x,y\in(0,\infty)\), we have

$$ 2\psi_{k} \biggl(\frac{x+y}{2} \biggr)- \psi_{k}(x)-\psi_{k}(y)\geq\ln \frac {(e^{\frac{1}{x}}-1)(e^{\frac{1}{y}}-1)}{(e^{\frac{2}{x+y}}-1)^{2}}. $$
(1.9)

Using the Theorems 1.1 and 1.2, we easily obtain the following Corollary 1.1.

Corollary 1.1

For \(0< k\leq1\) and \(x\in(0,\infty)\), we have

$$ \psi_{k}'(x)>\frac{1}{(1-e^{-\frac{1}{x}})x^{2}} $$
(1.10)

and

$$ \psi_{k}''(x)< \frac{e^{-\frac{1}{x}}-2x(1-e^{-\frac {1}{x}})}{(1-e^{-\frac {1}{x}})^{2}x^{4}}. $$
(1.11)

Theorem 1.3

For \(x>0\) and \(k\geq1\), we have

$$ \frac{\ln k-\gamma}{k}+x\psi_{k}' \biggl(k+ \frac{x}{2} \biggr)< \psi _{k}(x+k)< \frac{\ln k-\gamma}{k}+x \psi_{k}'\bigl(\sqrt{k(k+x)} \bigr). $$
(1.12)

Theorem 1.4

For \(p,k>0\) and every positive integer \(m\geq4\), the function

$$\phi_{m,p,k} (x) = \frac{{ [ {\psi_{p,k}^{(m)} (x)} ]^{4} }}{{\psi_{p,k}^{(m - 3)} (x)\psi_{p,k}^{(m - 1)} (x)\psi_{p,k}^{(m + 1)} (x) \psi_{p,k}^{(m +3)} (x)}} $$

is strictly decreasing on \((0,\infty)\) with

$$ \lim_{x \to\infty}\phi_{m,p,k} (x) = \frac {(m-3)(m-2)(m-1)^{2}}{{m^{2} (m+1)(m+2)}} $$
(1.13)

and

$$ \lim_{x \to0}\phi_{m,p,k} (x) = \frac{(m-2)(m-1) m^{2}}{{(m + 1)^{2}(m+2)(m+3)}}. $$
(1.14)

As a result, for \(p,k,x>0\) and every positive integer \(m\geq4\), we have

$$\begin{aligned} \frac{(m-3)(m-2)(m-1)^{2}}{{m^{2} (m+1)(m+2)}} &< \frac{{ [ {\psi_{p,k}^{(m)} (x)} ]^{4} }}{{\psi_{p,k}^{(m - 3)} (x)\psi_{p,k}^{(m - 1)} (x)\psi_{p,k}^{(m + 1)} (x) \psi _{p,k}^{(m +3)} (x)}} \\ &< \frac{(m-2)(m-1) m^{2}}{{(m + 1)^{2}(m+2)(m+3)}}. \end{aligned}$$

Theorem 1.5

For \(p,k,x>0\), the inequalities

$$ \frac{k}{{\ln ( {\frac{{B + 2k}}{{B + k}}} )}} < \psi _{p,k}^{ - 1} (x) < \frac{{k(p + 1)e^{kx} }}{{pk - e^{kx} }} + \frac{k}{2} $$
(1.15)

hold where \(B = \frac{{k(p + 1)e^{kx} }}{{pk - e^{kx} }}\).

Lemmas

Lemma 2.1

[42] If f is a function defined in an infinite interval I such that

$$ f(x)-f(x+\epsilon)>0\quad \textit{and}\quad \lim_{x\rightarrow\infty }f(x)= \delta $$

for some \(\epsilon>0\), then \(f(x)>\delta\) on I.

Remark 2.1

Lemma 2.1 was first proposed by Professor Feng Qi. It is simple, but has been validated in [15, 41, 42] to be especially effective in proving monotonicity and complete monotonicity of functions involving the gamma, psi and polygamma functions. The reader may refer to [40] and the references therein.

Lemma 2.2

For \(k>0\), the function \(\alpha(x)=[\psi_{k}'(x)]^{2}+\psi_{k}''(x)\) is positive on \((0,\infty)\) if and only if \(k\leq1\).

Proof

Direct computation yields

$$\begin{aligned} \alpha(x)-\alpha(x+k) &= \bigl[\psi_{k}'(x)- \psi_{k}'(x+k)\bigr] \bigl[\psi_{k}'(x)+ \psi_{k}'(x+k)\bigr]+\psi _{k}''(x)- \psi _{k}''(x+k) \\ &= \frac{2}{x^{2}} \biggl[\psi_{k}'(x)- \frac{1}{2x^{2}}-\frac{1}{x} \biggr] \\ &\triangleq\frac{2}{x^{2}}\beta(x), \end{aligned}$$

and

$$\begin{aligned} \beta(x+k)-\beta(x) &= \psi_{k}'(x+k)- \frac{1}{2(x+k)^{2}}-\frac{1}{x+k}-\psi_{k}'(x)+ \frac {1}{2x^{2}}+\frac{1}{x} \\ &= \frac{1}{x}-\frac{1}{2x^{2}}-\frac{1}{x+k}-\frac{1}{2(x+k)^{2}} \\ &= \frac{2(k-1)x^{2}+2k(k-1)x-k^{2}}{2x^{2}(x+k)^{2}}. \end{aligned}$$

It is easily observed that \(\beta(x+k)-\beta(x)<0\) if and only if \(k\leq1\). We complete the proof by using Lemma 2.1. □

Lemma 2.3

The following limit identity holds true:

$$ \lim_{x\rightarrow0^{+}} \biggl[\ln\bigl(e^{\frac{1}{x}}-1 \bigr)-\frac {1}{x} \biggr]=0. $$
(2.1)

Proof

By applying twice l’Hôspital rule, we easily complete the proof. □

Lemma 2.4

For \(k>0\), the inequalities

$$ \frac{1}{kx}\leq\psi_{k}'(x)\leq \frac{1}{kx}+\frac{1}{x^{2}} $$
(2.2)

hold true for any \(x\in(0,\infty)\).

Proof

Using the inequalities in [34], namely

$$ \frac{1}{k} \biggl(\frac{1}{x}-\frac{1}{x+pk+k} \biggr)\leq\psi _{p,k}'(x)\leq\frac{1}{k} \biggl( \frac{1}{x}-\frac{1}{x+pk+k} \biggr)+\frac {1}{x^{2}}- \frac{1}{(x+pk+k)^{2}}, $$
(2.3)

we easily obtain (2.2) as \(p\rightarrow+\infty\). □

Lemma 2.5

([25, 26, 43, 48])

Let \(\{a_{n}\}\) and \(\{b_{n}\}\) (\(n=0,1,2,\ldots\)) be real numbers such that \(b_{n}>0\) and \(\{\frac{a_{n}}{b_{n}}\}_{n\geq0}\) be increasing (resp., decreasing), then \(\{\frac{a_{0}+a_{1}+\cdots+a_{n}}{b_{0}+b_{1}+\cdots+b_{n}}\}\) is increasing (resp., decreasing).

Lemma 2.6

For \(p,k,x>0\) and every positive integer \(m\geq2\), the following limit identity holds true:

$$\lim_{x\rightarrow0^{+}} x^{m + 1}\psi_{p,k}^{(m)} (x) = \frac{{( - 1)^{m} (m - 1)!}}{k}. $$

Proof

Considering the inequalities (see [34, Theorem 2.7])

$$\frac{1}{k} \biggl( {\frac{1}{x} - \frac{1}{{x + pk + k}}} \biggr) \le \psi'_{p,k} (x) \le\frac{1}{k} \biggl( { \frac{1}{x} - \frac{1}{{x + pk + k}}} \biggr) + \frac{1}{{x^{2} }} - \frac{1}{{ ( {x + pk + k} )^{2} }} $$

and differentiating them \(m-1\) times, we easily complete the proof. □

Proofs of theorems

First proof of Theorem 1.1

A simple calculation gives

$$\begin{aligned} e^{\phi_{k}(x)}&=e^{\psi_{k}(x)}\bigl(e^{\frac{1}{x}}-1\bigr)=e^{\psi_{k}(x)+\frac {1}{x}}-e^{\psi_{k}(x)} \\ &=e^{\psi_{k}(x+k)}-e^{\psi_{k}(x)} \\ &\triangleq{\delta_{k}(x)} \end{aligned}$$

and

$$\begin{aligned} \delta_{k}'(x)&=e^{\psi_{k}(x+k)}\psi_{k}'(x)-e^{\psi_{k}(x)} \psi_{k}'(x) \\ &\triangleq\mu_{k}(x+k)-\mu_{k}(x). \end{aligned}$$

Using Lemma 2.2, we easily obtain

$$\mu_{k}'(x)=e^{\psi_{k}(x)}\bigl[\bigl( \psi_{k}'(x)\bigr)^{2}+\psi_{k}''(x) \bigr]>0. $$

This implies that the function \(\mu_{k}(x)\) is strictly increasing, and so \(\delta_{k}'(x)>0\) on \((0,\infty)\). As a result, the function \(e^{\phi _{k}(x)}\) is also strictly increasing on \((0,\infty)\). Considering Lemma 2.3, we have

$$ \lim_{x\rightarrow0^{+}}\phi_{k}(x)=\frac{\ln k-\gamma}{k}\quad \mbox{and}\quad \lim_{x\rightarrow\infty}\phi_{k}(x)=0. $$

The proof of Theorem 1.1 is completed. □

Second proof of Theorem 1.1

It is easily observed that \(\delta_{k}'(x)>0\) is equivalent to

$$ e^{\frac{1}{x}}\psi_{k}'(x+k)- \psi_{k}'(x)>0. $$
(3.1)

Considering Lemma 2.4, we only need to prove

$$ e^{\frac{1}{x}}\frac{1}{k(x+k)}>\frac{1}{kx}+ \frac{1}{x^{2}}. $$
(3.2)

Taking the logarithm to both sides of (3.2), we prove

$$ \frac{1}{x}+\ln\frac{1}{k}+\ln\frac{1}{x+k}>\ln \frac{x+k}{kx^{2}}. $$
(3.3)

So, we only need to prove

$$ \lambda_{k}(x)=\frac{1}{x}-\ln k-\ln{(x+k)}-\ln \frac{x+k}{kx^{2}}>0. $$
(3.4)

Since \(k\leq1\), we easily get

$$ \lambda_{k}'(x)=\frac{-2kx^{2}+(1-k)x+k(1-k)}{kx^{2}(x+k)}< 0. $$
(3.5)

This implies that the function \(\lambda_{k}(x)\) is strictly decreasing on \((0,\infty)\) with \(\lim_{x\rightarrow\infty}\lambda_{k}(x)=0\). Hence, we have \(\lambda_{k}(x)>0\). The proof is completed. □

Third proof of Theorem 1.1

Direct calculation results in

$$ \phi_{k}'(x)=\psi_{k}'(x)- \frac{e^{\frac{1}{x}}}{(e^{\frac{1}{x}}-1)^{2}x^{2}} $$
(3.6)

and

$$ \phi_{k}'(x)-\phi_{k}'(x+k)= \frac{1}{x^{2}}-\frac{e^{\frac {1}{x}}}{(e^{\frac {1}{x}}-1)x^{2}}+\frac{e^{\frac{1}{x+k}}}{(e^{\frac{1}{x+k}}-1)(x+k)^{2}} $$
(3.7)

with \(\lim_{x\rightarrow+\infty}\phi_{k}'(x)=0\).

In order to prove \(\phi_{k}'(x)-\phi_{k}'(x+k)>0\) for \(x>0\), it suffices to show

$$ x^{2}\bigl(e^{\frac{1}{x}}-1\bigr)>(x+k)^{2} \bigl(1-e^{-\frac{1}{x+k}} \bigr). $$
(3.8)

So, we only need to prove

$$ 1-k+\sum_{n=3}^{\infty} \frac{1}{n!} \biggl(\frac{1}{x^{n-2}}+\frac {(-1)^{n}}{(x+k)^{n-2}} \biggr)>0, $$
(3.9)

which is valid. By using Lemma 2.1, we can conclude that \(\phi _{k}'(x)>0\). Hence, the function \(\phi_{k}(x)\) is strictly increasing on \((0,\infty)\). □

Proof of Theorem 1.2

Using formula (3.7), we have

$$\begin{aligned} \phi_{k}''(x)-\phi_{k}''(x+k)&= \bigl(\phi_{k}'(x)-\phi_{k}'(x+k) \bigr)' \\ &=\frac{e^{\frac{1}{x+k}} [1+2k+2x-2(x+k)e^{\frac{1}{x+k}} ]}{ (e^{\frac{1}{x+k}}-1 )^{2}(x+k)^{4}} -\frac{(1-2x)e^{\frac{1}{x}}+2x}{(e^{\frac{1}{x}}-1)^{2}x^{4}}. \end{aligned}$$

For \(x>0\), the fact \(\phi_{k}''(x)-\phi_{k}''(x+k)<0\) is equivalent to

$$ \frac{(e^{\frac{1}{x}}-1 )^{2}}{(e^{\frac{1}{x+k}}-1 )^{2}}> \frac{(x+k)^{4}}{x^{4}}\frac{(1-2x)e^{\frac{1}{x}}+2x}{e^{\frac {1}{x+k}} [1+2k+2x-2(x+k)e^{\frac{1}{x+k}} ]}. $$
(3.10)

Applying inequality (3.8), we need to prove

$$ \triangle_{k}(x)=2k+1+2x-2ke^{\frac{1}{x+k}}-(1+2x)e^{\frac {1}{x}+\frac {1}{x+k}}< 0. $$
(3.11)

An easy calculation yields

$$\begin{aligned} \triangle_{k}'(x)&=\frac{ [4(1-k)x^{3}+(2+4k-2k^{2})x^{2}+(2k+2k^{2})x-2x^{4}+k^{2} ]e^{\frac {1}{x}+\frac {1}{x+k}}}{x^{2}(x+k)^{2}} \\ &\quad {}+\frac{2k e^{\frac{1}{x+k}}}{(x+k)^{2}}+2 \end{aligned}$$

and

$$ \triangle_{k}''(x)=\frac{q_{n}(x)e^{\frac{1}{x+k}}+r_{n}(x)e^{\frac {1}{x}+\frac{1}{x+k}}}{x^{4}(x+k)^{4}} $$

with \(\lim_{x\rightarrow\infty}\triangle_{k}'(x)=0\), where

$$ q_{n}(x)=-4kx^{5}-2k(1+2k)x^{4} $$

and

$$\begin{aligned} r_{n}(x) =&4(k-3)x^{5}+2\bigl(2k^{2}-13k-2 \bigr)x^{4}-4k(2+7k)x^{3} \\ &{}-8k^{2}(1+2k)x^{2}-4k^{3}(1+k)x-k^{4}. \end{aligned}$$

For \(0< k\leq1\), we easily obtain

$$ q_{n}(x)< 0, \qquad r_{n}(x)< 0. $$

This implies that \(\triangle_{k}'(x)\) is strictly decreasing and \(\triangle_{k}(x)\) is strictly increasing on \((0,\infty)\). Using \(\lim_{x\rightarrow\infty}\triangle_{k}(x)=-4<0\) and Lemma 2.1, we complete the proof. □

Proof of Theorem 1.3

Using (1.1) and (1.2), we get

$$ \psi_{k}(x+k)=\psi_{k}(x)+\frac{1}{x}= \frac{\ln k-\gamma}{k}+\sum_{n=1}^{\infty} \biggl( \frac{1}{nk}-\frac{1}{nk+x} \biggr). $$

By the mean value theorem for differentiation, there exists a number \(\sigma_{k,n}=\sigma_{k,n}(x)\) such that \(0<\sigma_{k,n}<x\) and

$$ \frac{1}{nk}-\frac{1}{nk+x}=\frac{x}{(nk+\sigma_{k,n})^{2}}. $$

Hence, we find

$$ \sigma_{k,n}=\sqrt{nk(nk+x)} -nk. $$

It is well known that the function \(\sigma_{k,n}\) is strictly increasing in k on \([1,+\infty)\) with

$$\begin{aligned}& \sigma_{k,1}=\sqrt{k(k+x)} -k, \\& \sigma_{k,\infty}=\lim_{n\rightarrow\infty}\sigma_{k,n}= \frac{x}{2}. \end{aligned}$$

Therefore, we get

$$ x\sum_{n=1}^{\infty}\frac{1}{(nk+\sigma_{k,\infty})^{2}}< \psi _{k}(x+k)-\frac {\ln k-\gamma}{k}< x\sum_{n=1}^{\infty}\frac{1}{(nk+\sigma_{k,1})^{2}}. $$

This completes the proof. □

Proof of Theorem 1.4

By (1.6) and direct computation, we have

$$\begin{aligned}& \frac{{ ( {\psi_{p,k}^{(m)} (x)} )^{4} }}{{\psi_{p,k}^{(m - 3)} (x)\psi_{p,k}^{(m - 1)} (x)\psi_{p,k}^{(m + 1)} (x)\psi _{p,k}^{(m + 3)} (x)}} \\& \quad = A\frac{{\sum_{n = 0}^{p} {\sum_{\lambda= 0}^{n} {\sum_{k = 0}^{\lambda}{\sum_{i = 0}^{n - \lambda} {\frac{1}{{(ik + x)^{2m + 2} ((n - i)k + x)^{2m + 2} }}} } } } }}{{\sum_{n = 0}^{p} {\sum_{\lambda = 0}^{n} {\sum_{k = 0}^{\lambda}{\sum_{i = 0}^{n - \lambda} {\frac {1}{{(ik + x)^{2m - 2} ((n - i)k + x)^{2m + 4} }}} } } } }}, \end{aligned}$$

where \(A = \frac{{ ( {m!} )^{4} }}{{(m - 3)!(m - 1)!(m + 1)!(m + 3)!}}\). Let us define sequences \(\{ {\alpha_{m,i} } \}_{i \ge0} \), \(\{ {\beta_{m,i} } \}_{i \ge0} \) and \(\{ {\omega_{m,i} } \}_{i \ge0} \) by

$$\begin{aligned}& \alpha_{m,i} = \frac{1}{{ ( {ik + x} )^{2m + 2} [ { ( {n - i} )k + x} ]^{2m + 2} }}, \\& \beta_{m,i} = \frac{1}{{ ( {ik + x} )^{2m-2} [ { ( {n - i} )k + x} ]^{2m +4} }}, \end{aligned}$$

and

$$\omega_{m,i} =\frac{{\alpha_{m,i} }}{{\beta_{m,i} }} = \biggl(\frac {{ ( {n - i} )k + x}}{{ik + x}} \biggr)^{4}. $$

It follows that

$$\frac{{\omega_{m,i + 1} }}{{\omega_{m,i} }} = \biggl(\frac{{ [ { ( {n - i - 1} )k + x} ] ( {ik + x} )}}{{ [ { ( {i + 1} )k + x} ] [ { ( {n - i} )k + x} ]}} \biggr)^{4}. $$

It is not difficult to see that the fact \(\frac{{\omega_{m,i + 1} }}{{\omega_{m,i} }} < 1 \) is equivalent to

$$\begin{aligned}& \bigl[ { ( {n - i - 1} )k + x} \bigr] ( {ik + x} ) < \bigl[ { ( {i + 1} )k + x} \bigr] \bigl[ { ( {n - i} )k + x} \bigr] \\& \quad \Leftrightarrow\quad - nk^{2} - 2kx < 0. \end{aligned}$$

So the sequence \(\{ {\omega_{m,i} } \}_{i \ge0} \) is strictly decreasing. This implies that the function \(\phi_{m,p,k} ( x ) \) is strictly decreasing on \((0,\infty)\) by Lemma 2.5. From the identity

$$\psi_{p,k}^{ ( m )} ( {x + k} ) = ( { - 1} )^{m} \frac{{m!}}{{x^{m + 1} }} - ( { - 1} )^{m} \frac {{m!}}{{ ( {x + pk + k} )^{m + 1} }} + \psi_{p,k}^{ ( m )} ( x ), $$

we easily obtain (1.14). Using Lemma 2.6, we get (1.13). This completes the proof. □

Proof of Theorem 1.5

Using (1.4) and the functional equation (see [35])

$$\Gamma_{p,k} (x + k) = \frac{{pkx}}{{x + pk + k}}\Gamma_{p,k} (x), $$

we obtain, after a direct computation, that

$$\begin{aligned}& \ln\Gamma_{p,k} (x + k) = \ln(p + 1)! + (p + 1)\ln k + \biggl( {\frac {{x + k}}{k} - 1} \biggr)\ln pk - \sum _{i = 0}^{p} {\ln \bigl( {x + ( {i + 1} )k} \bigr)}, \end{aligned}$$
(3.12)
$$\begin{aligned}& \ln\Gamma_{p,k} (x) = \ln(p + 1)! + (p + 1)\ln k + \biggl( {\frac {x}{k} - 1} \biggr)\ln pk - \sum _{i = 0}^{p} {\ln ( {x + ik} )}, \end{aligned}$$
(3.13)

and

$$ \ln\Gamma_{p,k} (x + k) = \ln\frac{{pkx}}{{x + pk + k}} + \ln \Gamma _{p,k} (x). $$
(3.14)

Combining (3.12) and (3.13) with (3.14), we get

$$ \ln\frac{{pkx}}{{x + pk + k}} = \ln pk - \sum _{i = 0}^{p} {\ln\frac{{x + ( {i + 1} )k}}{{x + ik}}.} $$
(3.15)

By the mean value theorem, we obtain

$$ \ln\frac{{x + ( {i + 1} )k}}{{x + ik}} = \frac{k}{{ik + \rho(i)}},\quad \rho(i) \in(x,x + k). $$
(3.16)

Hence, identity (3.15) changes into

$$ \ln\frac{{pkx}}{{x + pk + k}} = k \Biggl( {\frac{1}{k}\ln pk - \sum _{i = 0}^{p} {\ln\frac{1}{{ik + \rho(i)}}} } \Biggr). $$
(3.17)

From identity (3.16), we conclude that

$$\rho(i) = \frac{k}{{\ln ( {1 + \frac{k}{{x + ik}}} )}} - ik. $$

Next, we show that ρ is strictly increasing on \((1,\infty)\). Differentiating \(\rho(i)\), we observe that \(\rho'(i)>0\) if and only if

$$\sqrt{(x + ik) (x + ik + k)} < \frac{{(x + ik + k) - (x + ik)}}{{\ln(x + ik + k) - \ln(x + ik)}}, $$

which follows from the geometric–logarithmic mean inequality. A simple computation yields \(\rho(1) = \frac{k}{{\ln ( {\frac{{x + 2k}}{{x + k}}} )}} - k \) and \(\rho(\infty) = \lim_{i \to\infty} \rho(i) = x + \frac{k}{2}\). Since \(\psi_{p,k}\) and \(\psi^{-1}_{p,k}\) are strictly increasing on \((0,\infty)\), we easily obtain that

$$\psi_{p,k} \bigl( {\rho(1)} \bigr) < \frac{1}{k}\ln \frac{{pkx}}{{x + pk + k}} < \psi_{p,k} \bigl( {\rho(\infty)} \bigr). $$

Hence we have

$$\frac{k}{{\ln ( {\frac{{x + 2k}}{{x + k}}} )}} - k < \psi _{p,k}^{ - 1} \biggl( { \frac{1}{k}\ln\frac{{pkx}}{{x + pk + k}}} \biggr) < x + \frac{k}{2}. $$

Replacing x by \(\frac{{k(p + 1)e^{kx} }}{{pk - e^{kx} }} \) here completes the proof. □

A conjecture

Finally, we give a conjecture.

Conjecture 4.1

For \(p>0\) and \(k\geq1\), the function

$$ \phi_{p,k}(x)=\psi_{p,k}(x)+\ln \bigl(e^{\frac{1}{x}-\frac {1}{x+pk+k}}-1 \bigr) $$

is strictly decreasing from \((0,\infty)\) onto \((-\infty,\psi_{p,k}(k))\).

Remark 4.1

It is natural to ask whether the monotonicity result of Theorem 1.1 can be extended to the digamma function \(\psi_{p,k}(x)\) with two parameters by using the method of Theorem 1.1. Unfortunately, we failed to prove Conjecture 4.1. Alzer’s work shows that the function \(\phi_{k}(x)=\psi_{k}(x)+\ln (e^{\frac {1}{x}}-1)\) is useful for studying harmonic numbers. This is related to the formula (see [35, Remark 2.1])

$$\phi_{p,k}(k)=\frac{1}{k}\bigl[\ln(pk)-H(p+1)\bigr], $$

where \(H(n)\) is the nth harmonic number. So, it would be a meaningful result if anyone can prove this conjecture.

Remark 4.2

The \((p,k)\)-generalized Nielsen’s β-function can be defined as

$$\begin{aligned} \beta_{p,k}(x)&= \int_{0}^{1} \frac{1-t^{2k(p+1)}}{1+t^{k}}t^{x-1}\,dt \\ &= \int_{0}^{\infty} \frac{1-e^{-2k(p+1)t}}{1+e^{-kt}} e^{-xt} \,dt \\ &=\sum^{p}_{n=0} \biggl( \frac{1}{2nk+x}-\frac{1}{2nk+k+x} \biggr) \\ &=\frac{1}{2} \biggl\{ \psi_{p,k} \biggl(\frac{x+k}{2} \biggr)-\psi_{p,k} \biggl(\frac{x}{2} \biggr) \biggr\} , \end{aligned}$$

where \(k\in(0,1]\), \(p, x \in(0,\infty)\), and \(\lim_{p\rightarrow \infty }\beta_{p,k}(x)=\beta_{k}(x)\). Analogously to Remark 1.1, if Conjecture 4.1 holds true, we can estimate the upper and lower bounds of this function \(\beta_{p,k}(x)\).

Results and discussion

Some monotonicity and concavity properties of the k and \((p,k)\)-analogues of the digamma and polygamma functions were deeply studied. In doing so, we established some inequalities involving the generalized digamma and polygamma functions. Theorems 1.11.3 are extensions of some known results. Theorem 1.4 is not only a completely new result, it’s even new for \(\psi (x)\). In addition, the method of proof is also new. Theorem 1.5 gives an inequality for the inverse of the digamma function. At the moment, such results are very few. In the end, we stated a conjecture involving the \((p,k)\)-analogue of the digamma function.

Methods and experiment

Not applicable.

References

  1. Abramowitz, M., Stegun, I. (eds.): Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Dover, New York (1965)

    MATH  Google Scholar 

  2. Alzer, H.: On some inequalities for the gamma and psi function. Math. Comput. 66, 373–389 (1997)

    MathSciNet  Article  Google Scholar 

  3. Alzer, H.: Sharp inequalities for the digamma and polygamma functions. Forum Math. 16, 181–221 (2004)

    MathSciNet  Article  Google Scholar 

  4. Alzer, H.: Sharp inequalities for the harmonic numbers. Expo. Math. 24(4), 385–388 (2006)

    MathSciNet  Article  Google Scholar 

  5. Batir, N.: Some new inequalities for gamma and polygamma functions. J. Inequal. Pure Appl. Math. 6(4), Article ID 103 (2005)

    MathSciNet  MATH  Google Scholar 

  6. Batir, N.: On some properties of digamma and polygamma functions. J. Math. Anal. Appl. 328(1), 452–465 (2014)

    MathSciNet  Article  Google Scholar 

  7. Boyadzhiev, K.N., Medina, L.A., Moll, V.H.: The integrals in Gradshteyn and Ryzhik, part II: the incomplete beta function. Scientia, Ser. A, Math. Sci. 18, 61–75 (2009)

    MathSciNet  MATH  Google Scholar 

  8. Chen, Y.-C., Mansour, T., Zou, Q.: On the complete monotonicity of quotient of gamma functions. Math. Inequal. Appl. 15(2), 395–402 (2012)

    MathSciNet  MATH  Google Scholar 

  9. Chiu, S.N., Yin, C.-C.: On the complete monotonicity of the compound geometric convolution with applications to risk theory. Scand. Actuar. J. 2014(2), 116–124 (2014)

    MathSciNet  Article  Google Scholar 

  10. Coffey, M.C.: On one-dimensional digamma and polygamma series related to the evaluation of Feynman diagrams. J. Comput. Appl. Math. 183, 84–100 (2005)

    MathSciNet  Article  Google Scholar 

  11. Díaz, R., Pariguan, E.: On hypergeometric functions and Pochhammer k-symbol. Divulg. Mat. 15(2), 179–192 (2007)

    MathSciNet  MATH  Google Scholar 

  12. Dong, H., Yin, C.-C.: Complete monotonicity of the probability of ruin and DE Finetti’s dividend problem. J. Syst. Sci. Complex. 25(1), 178–185 (2012)

    MathSciNet  Article  Google Scholar 

  13. Guan, Y.-L., Zhao, Z.-Q., Lin, X.-L.: On the existence of positive solutions and negative solutions of singular fractional differential equations via global bifurcation techniques. Bound. Value Probl. 2016, 141 (2016)

    MathSciNet  Article  Google Scholar 

  14. Guo, B.-N., Qi, F.: Some properties of the psi and polygamma functions. Hacet. J. Math. Stat. 39(2), 219–231 (2010)

    MathSciNet  MATH  Google Scholar 

  15. Guo, B.-N., Qi, F.: Two new proofs of the complete monotonicity of a function involving the psi function. Bull. Korean Math. Soc. 47(1), 103–111 (2010)

    MathSciNet  Article  Google Scholar 

  16. Guo, B.-N., Qi, F., Srivastava, H.M.: Some uniqueness results for the non-trivially complete monotonicity of a class of functions involving the polygamma and related functions. Integral Transforms Spec. Funct. 21(11), 849–858 (2010)

    MathSciNet  Article  Google Scholar 

  17. Guo, B.-N., Zhao, J.-L., Qi, F.: A completely monotonic function involving divided differences of the tri- and tetra-gamma functions. Math. Slovaca 63(3), 469–478 (2013)

    MathSciNet  Article  Google Scholar 

  18. Guo, Y.-X.: Solvability for a nonlinear fractional differential equation. Bull. Aust. Math. Soc. 80, 125–138 (2009)

    MathSciNet  Article  Google Scholar 

  19. Guo, Y.-X.: Nontrivial solutions for boundary-value problems of nonlinear fractional differential equations. Bull. Korean Math. Soc. 47, 81–87 (2010)

    MathSciNet  Article  Google Scholar 

  20. Jiang, J.-Q., Liu, L.-S.: Existence of solutions for a sequential fractional differential system with coupled boundary conditions. Bound. Value Probl. 2016, 159 (2016)

    MathSciNet  Article  Google Scholar 

  21. Krasniqi, F., Shabani, A.: Convexity properties and inequalities for a generalized gamma functions. Appl. Math. E-Notes 10, 27–35 (2010)

    MathSciNet  MATH  Google Scholar 

  22. Krasniqi, V., Mansour, T., Shabani, A.S.: Some monotonicity properties and inequalities for Γ and ζ functions. Math. Commun. 15(2), 365–376 (2010)

    MathSciNet  MATH  Google Scholar 

  23. Krasniqi, V., Mansour, T., Shabani, A.S.: Some inequalities for q-polygamma function and \(\zeta _{q}\)-Riemann zeta functions. Ann. Math. Inform. 37, 95–100 (2010)

    MathSciNet  MATH  Google Scholar 

  24. Lin, X.-L., Zhao, Z.-Q.: Iterative technique for a third-order differential equation with three-point nonlinear boundary value conditions. Electron. J. Qual. Theory Differ. Equ. 2016, 12 (2016)

    MathSciNet  Article  Google Scholar 

  25. Mehrez, M., Sitnik, S.M.: Proofs of some conjectures on monotonicity of ratios of Kummer, Gauss and generalized hypergeometric functions. Analysis 36(4), 263–268 (2016). http://arxiv.org/abs/1411.6120

    MathSciNet  MATH  Google Scholar 

  26. Mehrez, M., Sitnik, S.M.: Monotonicity of ratios of q-Kummer confluent hypergeometric and q-hypergeometric functions and associated Turán types inequalities. Mat. Vesn. 68(3), 225–231 (2016)

    MATH  Google Scholar 

  27. Merkle, M.: Inequalities for the gamma function via convexity. In: Cerone, P., Dragomir, S.S. (eds.) Advances in Inequalities for Special Functions, pp. 81–100. Nova Science Publishers, New York (2008)

    Google Scholar 

  28. Mortici, C.: A quicker convergence toward the gamma constant with the logarithm term involving the constant e. Carpath. J. Math. 26(1), 86–91 (2010)

    MATH  Google Scholar 

  29. Mortici, C.: Very accurate estimates of the polygamma functions. Asymptot. Anal. 68(3), 125–134 (2010)

    MathSciNet  MATH  Google Scholar 

  30. Nantomah, K.: Convexity properties and inequalities concerning the \((p,k)\)-gamma functions. Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat. 66(2), 130–140 (2017)

    MathSciNet  MATH  Google Scholar 

  31. Nantomah, K.: Monotoicity and convexity properties of the Nielsen’s β-function. Probl. Anal. Issues Anal. 6(24)(2), 81–93 (2017)

    MathSciNet  Article  Google Scholar 

  32. Nantomah, K.: Monotoicity and convexity properties and some inequalities involving a generalized form of the Wallis’ cosine formula. Asian Res. J. Math. 6(3), 1–10 (2017)

    MathSciNet  Article  Google Scholar 

  33. Nantomah, K.: A generalization of the Nielsen’s β-function. Int. J. Open Probl. Comput. Sci. Math. 11(2), 16–26 (2018)

    MathSciNet  Google Scholar 

  34. Nantomah, K., Merovci, F., Nasiru, S.: Some monotonic properties and inequalities for the \((p,q)\)-gamma function. Kragujev. J. Math. 42(2), 287–297 (2018)

    Article  Google Scholar 

  35. Nantomah, K., Prempeh, E., Twum, S.B.: On a \((p,k)\)-analogue of the gamma function and some associated inequalities. Moroccan J. Pure Appl. Anal. 2(2), 79–90 (2016)

    MATH  Google Scholar 

  36. Nielsen, N.: Handbuch der Theorie der Gamma funktion, 1st edn. Teubner, Leipzig (1906)

    MATH  Google Scholar 

  37. Qi, F.: Bounds for the ratio of two gamma functions. J. Inequal. Appl. 2010, Article ID 493058 (2010)

    MathSciNet  Article  Google Scholar 

  38. Qi, F., Cui, R.-Q., Chen, C.-P., Guo, B.-N.: Some completely monotonic functions involving polygamma functions and an application. J. Math. Anal. Appl. 310(1), 303–308 (2005)

    MathSciNet  Article  Google Scholar 

  39. Qi, F., Guo, B.-N.: Completely monotonic functions involving divided differences of the di- and tri-gamma functions and some applications. Commun. Pure Appl. Anal. 8(6), 1975–1989 (2009)

    MathSciNet  Article  Google Scholar 

  40. Qi, F., Guo, B.-N.: Necessary and sufficient conditions for functions involving the tri- and tetra-gamma functions to be completely monotonic. Adv. Appl. Math. 44(1), 71–83 (2010)

    MathSciNet  Article  Google Scholar 

  41. Qi, F., Guo, B.-N.: A class of completely monotonic functions involving divided differences of the psi and tri-gamma functions and some applications. J. Korean Math. Soc. 48(3), 655–667 (2011)

    MathSciNet  Article  Google Scholar 

  42. Qi, F., Guo, S.-L., Guo, B.-N.: Completely monotonicity of some functions involving polygamma functions. J. Comput. Appl. Math. 233, 2149–2160 (2010)

    MathSciNet  Article  Google Scholar 

  43. Sitnik, S.M., Mehrez, M.: On monotonicity of ratios of some hypergeometric functions. Sib. Èlektron. Mat. Izv. 13, 260–268 (2016)

    MathSciNet  MATH  Google Scholar 

  44. Wang, Y., Liu, L.-S., Wu, Y.-H.: Positive solutions for a nonlocal fractional differential equation. Nonlinear Anal. 74(11), 3599–3605 (2011)

    MathSciNet  Article  Google Scholar 

  45. Wang, Y., Liu, L.-S., Wu, Y.-H.: Existence and uniqueness of a positive solution to singular fractional differential equations. Bound. Value Probl. 2012, 81 (2012)

    MathSciNet  Article  Google Scholar 

  46. Wang, Y., Liu, L.-S., Wu, Y.-H.: Positive solutions for a class of higher-order singular semipositone fractional differential system. Adv. Differ. Equ. 2014, 268 (2014)

    Article  Google Scholar 

  47. Xu, R., Meng, F.-W.: Some new weakly singular integral inequalities and their applications to fractional differential equations. J. Inequal. Appl. 2016, 78 (2016)

    MathSciNet  Article  Google Scholar 

  48. Yin, L., Cui, W.-Y.: A generalization of Alzer inequality related to exponential function. Proc. Jangjeon Math. Sci. 18(3), 385–388 (2016)

    MathSciNet  MATH  Google Scholar 

  49. Zheng, Z.-W., Zhang, X.-J., Shao, J.: Existence for certain systems of nonlinear fractional differential equations. J. Appl. Math. 2014, Article ID 376924 (2014)

    Google Scholar 

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Acknowledgements

The authors are grateful to anonymous referees and the editor for their careful corrections and valuable comments on the original version of this paper.

Funding

This work was supported by National Natural Science Foundation of China (Grant Nos. 11701320 and 11705122), the Science and Technology Foundations of Shandong Province (Grant Nos. J16li52, J14li54 and J17KA161) and Science Foundations of Binzhou University (Grant Nos. BZXYL1104 and BZXYL1704).

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Yin, L., Huang, LG., Song, ZM. et al. Some monotonicity properties and inequalities for the generalized digamma and polygamma functions. J Inequal Appl 2018, 249 (2018). https://doi.org/10.1186/s13660-018-1844-2

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MSC

  • 33B15
  • 26A48
  • 26A51

Keywords

  • Generalized digamma and polygamma functions
  • Monotonicity
  • Inequalities
  • Concavity